Two years ago, Amy Greenspon and Mitchell Algus moved in together. She was a blond, 31-year-old gallery director who looked a little like Alicia Silverstone in Clueless and regularly appeared in the party pages of Vogue and Style.com. He was an avuncular, opinionated 56-year-old science teacher at a public high school in Queens, who was also an art dealer. They were both children of doctors—her father was a wealthy Manhattan psychiatrist and art collector, his an upper-middle-class Long Island dentist.

They had their differences. He was into restoration; she preferred new. She wanted to keep the original façade and floors, he wanted to redo them. He won that battle.

But Ms. Greenspon has won others. The two are business partners, not romantic partners, and Algus Greenspon, their West Village art gallery, has the smooth cement floors he wanted but also shows some of the young artists she prefers.

Even by art world standards, it is a peculiar arrangement. What has made it work?

The two first spoke around 2000, when Ms. Greenspon was at NYU and working for the PaceWildenstein gallery. She contacted Mr. Algus because she was interested in buying a painting by the obscure Pop artist Nicholas Krushenick, whom he had shown. “I remember calling, and someone answered,” she said. “They were like, ‘Mitchell doesn’t come in until after 4.’ I thought, ‘O.K.. This guy must be really important.’”

A lot of people—high-profile artists, curators and writers—really do hold Mr. Algus in high regard. But the real reason he wasn’t at the gallery was that he was at Long Island City High School.

Mr. Krushenick was not the only obscure artist Mr. Algus showed. There were also Larry Zox, Harold Stevenson, Rosalyn Drexler and dozens more. From 1992 until he joined with Ms. Greenspon, he specialized almost exclusively in the work of unknown or forgotten artists from the 1960s and ’70s. He was heralded for reviving careers, sometimes to his own detriment.

“It’s the ones like Lee Lozano who get upsetting because you realize how powerless you are,” he said on a recent late afternoon over iced coffees at the Eli’s Essentials café next to the Whitney Museum. He’d just finished teaching. One of the leading conceptual artists in New York in the 1960’s, Ms. Lozano quit the art world and was virtually unknown when he and other dealers presented her work in 1998, shortly before her death. Museum retrospectives followed those shows, and international powerhouse Hauser & Wirth now works with her estate.

“We were trying to sell a painting for $18,000 that would go for over a million now,” he said. “Money is sexy. A painting that is a million dollars is a lot more sexy than a painting that is $18,000. It’s very frustrating. You realize that is the way the art world is.”

An article in this month’sForbes discusses today’s high-flying art market and talks about dealers being celebrities. Most aren’t.

“The art market,” Mr. Algus said at Eli’s, “has never meant anything to me.” Dealers often say this, but in Mr. Algus’s case it appears to be true. Over two decades he has run four relatively quiet galleries in four different neighborhoods, while by his own admission selling very little.

Selling was not really the point. It was about getting under people’s skin, aesthetically speaking. “I really dislike consensus,” he said. He propped his glasses on his head and continued emphatically, “When everybody agrees on something I just feel like it’s all the nasty kids I hated in high school, and so, if things can undermine consensus, that’s sort of what I’m interested in.”

That would be South Side Senior High School, in Rockville Centre on Long Island. “He had these swept bangs,” recalled artist Deborah Kass, a fellow student. “He was fucking adorable.” Mr. Algus is pretty much bald today, but still radiates boyish enthusiasm. At Eli’s, The Observer admitted unfamiliarity with the painter Louis Eilshemius. “You don’t know Louis Eilshemius?” He gaped in horror, then pulled up JPEGS on his laptop.

Art wasn’t originally in the cards for Mr. Algus. Though his father bought art magazines and took him to galleries, he studied geology at Harpur College, now SUNY Binghampton, and got his master’s in the subject at McGill University in Montreal. “I really wanted to go up to the Canadian Arctic. That whole British ‘to the ends of the Earth’ tradition was really alive there.”

He went to the University of Colorado-Boulder to get his doctorate and found himself hanging out in the art history department. But, he said, “It was very pretentious. It was really intellectually rigorous and very dry.” He went back to McGill and got his Ph.D. in geology.

In 1986, he moved back to New York to teach grad school, and he and a friend opened a gallery in Williamsburg. He was also making art—small assemblages he exhibited at East Village galleries.

Williamsburg was uncharted territory. “We would have an opening and there would be 500 people out in the street,” Mr. Algus said. They showed their friends’ work. “We would have a bonfire, and then no one would come again. It was just a party, and then a month of nothing.”

One day Mr. Algus’s girlfriend, Dyanne (who is now his wife), had her wallet stolen from her car. A man who’d done odd jobs for the gallery learned the thieves wanted $40 for it. “Then I get a phone call that he got into a fight with somebody and stabbed him and he needed a plane ticket to Puerto Rico and he was living in the basement hiding out from the police. So, the gallery was over.”

His next gallery, which he opened about nine months later, in November 1992, was on Thompson Street in Soho. There he began dusting off artists he’d read about in his father’s old art magazines. He stopped teaching graduate school and moved to a high school so that he could head to the gallery in the late afternoon.

He opened with The New Adam, a 39-foot-long nude portrait of Sal Mineo that painter Harold Stevenson—a well-regarded but then-unknown artist—made in 1962. The painting, which was subsequently donated to the Guggenheim, took up his whole gallery.

“He was like a fairy godmother for us,” Betty Tompkins said in her Soho loft recently, surrounded by a few of her “Fuck Paintings”—huge, virtuosic black-and-white airbrushed canvases of heterosexual penetration—that Mr. Algus showed, ending a decades-long drought in her career, which subsequently took off. Her first “Fuck Painting,” from 1969, is now owned by Paris’s Centre Pompidou.

Unlike at many galleries, women were integral to Mr. Algus’s program. There were Ms. Lozano and Ms. Tompkins, but also Joan Semmel, who paints fleshy full-body portraits, often in lurid colors, and Judith Bernstein, who’s known for brawny drawings of flying penises that sometimes cover entire walls.

All of those artists were mid-career or well past it when Mr. Algus entered the picture, and all now work with other galleries. When his artists moved on, they generally did so with Mr. Algus’s blessing. “It’s good, because I don’t want to represent people in that way,” he said. “It takes the pressure off me.”

With a small space, low overheard and few employees, he could be discerning in how he operated. “I was actually there at the gallery one time, and he’s on the phone with somebody—a collector, I think,” Ms. Tompkins said. She recalled him saying firmly, “No, that’s not what I do. I teach school so I can have the privilege of saying no to things like this.’”